A stagnant dispersive zone represents another type of immobile storage zone (in addition to a matrix diffusion zone) that can act to disperse solute transport in a Pipe.
The stagnant dispersive zone was designed to be used to represent the low-velocity or “stagnant” pores or regions within the plane of a fracture, or within a porous medium. Such zones are likely to exist in many types of subsurface systems. For example, for an aquifer under a uniform head gradient, the seepage velocity varies with the square of pore aperture, so in a medium with variable pore sizes it is possible for smaller pores to have a very low velocity, and hence act as a sort of storage zone.
For this kind of storage zone, interchange between the mobile zone and the storage zone may actually occur by both advective and diffusive processes. For relatively high gradient, short-term systems (such as those associated with tracer tests), the advective interchange will typically dominate the system, whereas for longer time periods under lower gradients the diffusive exchange process may dominate. (As represented in GoldSim, however, only advective exchange between the stagnant and the mobile zone is simulated.)
Note that it is actually this type of physical model that is represented when a classic “dispersivity” factor is used to describe solute dispersion. In reality, there is a continuum of velocities within a pathway (i.e., a continuum of “dispersive storage zones”). The classic dispersivity parameter is simply a factor that corrects for the fact that we typically model flow at a macro-scale, rather than a micro-scale (in which all of the dispersive zones are explicitly represented).
GoldSim allows the user to explicitly represent a single stagnant dispersive zone in a one-dimensional pathway. The user must specify the fraction of the pathway which is assumed to be “stagnant”. As noted by its name, this portion of the pathway is assumed to have negligible advective velocity. It can be filled with a porous medium (to which species can sorb) , which may be different from that in the mobile zone of the pathway (i.e., the infill medium). Transfer between the stagnant zone and the mobile zone is advective, and thus varies proportionately to the quantity of fluid flowing through the pathway. The constant of proportionality (referred to herein as the transfer rate), has dimensions of inverse length, and is defined as the probability of an individual solute molecule moving from the mobile zone to the stagnant zone, per length of distance traveled in the mobile zone.
Because the transfer rate is proportional to the flow rate in the pathway, this type of mobile-storage zone transfer is qualitatively different from transfer between the mobile zone and matrix diffusion zones. A stagnant dispersive zone acts to disperse solute flowing in the pathway in a unique manner. In the limit where the product of the transfer rate and the pathway length is much greater than one, a stagnant dispersive zone produces a dispersion profile identical to that associated with the conventional advection-dispersion equation. However, if this product is not significantly greater than one, it produces a highly asymmetric dispersion profile, characterized by a sharp leading edge and an elongated tail in the breakthrough curve (similar to that resulting from the presence of matrix diffusion zones).
The figure below illustrates the effect of a stagnant dispersive zone on the breakthrough of a slug input (in this case, the product of the transfer rate and pathway length is less than 1).
Note that if the total area and the flow rate of the pathway is held constant while the fraction of the pathway which is stagnant is increased (as in the figure shown above), the mean breakthrough curve is actually shifted to the left (shorter times), since the effective area of the mobile zone is decreased, and hence the velocity in the path is increased.